Biconditional Statements & Good Definitions
When both a conditional and its converse are true, we combine them into a single powerful statement: “if and only if.”
Introduction
A good geometric definition works in bothdirections. When you can say “if and only if,” you've written a biconditional — the gold standard for definitions.
Past Knowledge
Conditionals (2.1.3). Converse (2.1.4).
Today's Goal
Write biconditional statements and identify good definitions.
Future Success
Two-column proofs (2.3) use definitions as reasons — and every definition is biconditional.
Key Concepts
Biconditional Statement
A biconditional combines a conditional and its converse:
Read as: “ if and only if .”
Good Definitions
A statement is a good definition if it can be written as a true biconditional — both the conditional and its converse must be true.
Abbreviation
“If and only if” is often abbreviated iff in mathematics.
Worked Examples
Writing a Biconditional
Conditional: “If an angle is 90°, then it is a right angle.” Can this be a biconditional?
Converse: “If an angle is a right angle, then it is 90°.” — also true!
Biconditional: “An angle is 90° if and only if it is a right angle.”
Not a Good Definition
“If a figure is a square, then it has four sides.” Is this a good definition of a square?
Converse: “If a figure has four sides, then it is a square.” — false (rectangles, trapezoids have 4 sides too).
Answer: No — the converse is false, so this is NOT a good definition.
Common Pitfalls
Forgetting to Check BOTH Directions
A biconditional requires both the conditional AND the converse to be true. Checking only one direction is not enough.
Real-Life Applications
Password Systems
“You gain access if and only if the password is correct.” This is a biconditional — correct password ↔ access granted. If either direction fails (access without password, or no access with correct password), the system is broken.
Practice Quiz
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